Naturality of Rieffel's Morita equivalence for proper actions (Q409236)

In [``Integrable and proper actions on \(C^*\)-algebras, and square-integrable representations of groups'', Expo. Math. 22, No. 1, 1--53 (2004; Zbl 1047.22004)], \textit{M. A. Rieffel} proved that when \(\alpha\) is an action of a locally compact group \(G\) on a \(C^*\)-algebra \(A\) such that \(G\) acts freely and properly on a locally compact space \(T\) and there is a nondegenerate homomorphism from \(C_0(T)\to M(A)\) intertwining the right translation action of \(G\) and the extension of \(\alpha\), the action \(\alpha\) is \textit{proper} as in [\textit{M. A. Rieffel}, Prog. Math. 84, 141--182 (1991; Zbl 0749.22003)]. As a consequence, the reduced crossed product is Morita equivalent to a generalised fixed-point algebra \(\text{Fix}(A,\alpha)\) in \(M(A)\). In the present paper, the authors show that the assignment \((A,\alpha)\mapsto\text{Fix}(A,\alpha)\) extends to a functor \(\text{Fix}\) on an appropriate category in such a way that Rieffel's Morita equivalence implements a natural isomorphism between a crossed product functor and this new functor. The key technical steps require finding an appropriate analogue in this framework of a comma category as used by \textit{S. Kaliszewski} and \textit{J. Quigg} [``Categorical Landstad duality for actions'', Indiana Univ. Math. J. 58, No. 1, 415--442 (2009; Zbl 1175.46060)], and \textit{S. Kaliszewski, J. Quigg} and \textit{I. Raeburn} [``Proper actions, fixed-point algebras and naturality in nonabelian duality'', J. Funct. Anal. 254, No. 12, 2949--2968 (2008; Zbl 1151.46051)].